Initial Data U0 Research Articles

On the global solvability of porous media type equations with space dependent advection flux

We show that the Cauchy problem for advection-diffusion equations with nonlinear diffusion term div(|u|α∇u) and advection flux f(x,t,u) of order O(|u|1+k) for large u is globally solvable for arbitrary initial data u0∈L1(Rn)∩L∞(Rn) when k∈[0,α+1/n). Some important related results are also given.

Existence and supercontractive estimates for parabolic–elliptic systems

In this paper we study existence and summability of the solutions of the following parabolic–elliptic system of partial differential equations ut−div(A(x,t)∇u)=−div(uM(x)∇ψ)in Ω×(0,T), −div(M(x)∇ψ)=|u|θin Ω×(0,T), ψ(x,t)=0on ∂Ω×(0,T), u(x,t)=0on ∂Ω×(0,T), u(x,0)=u0(x)in Ω where θ∈(0,1), Ω is a bounded subset of RN, N>2, and T>0. We will prove existence results for initial data u0 in L1(Ω). Moreover, despite the datum u0 is assumed to be only a summable function and although the function uM(x)∇ψ in the divergence term of the first equation is not regular enough, there exist solutions that immediately improve their summability and belong to every Lebesgue space. Finally, we study the behavior in time of such solutions and we prove estimates that describe their blow-up for t near zero.

Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension

This manuscript considers the no-flux initial-boundary problem for the migration-consumption system{ut=Δ(uϕ(v)),vt=Δv−uv,(⋆) in a smoothly bounded domain Ω⊂Rn, n≥1, where ϕ suitably generalizes the singular prototype given byϕ(ξ)=ξ−α,ξ>0, with α>0.It is firstly shown that for such diffusion singularities of arbitrary strength, and for any given initial data u0 and v0 from W1,∞(Ω) with u0≥0 and v0>0 in Ω‾, a so-called very weak-strong solution (u,v) with (u,v)|t=0=(u0,v0) can be constructed. Under the additional restrictions that 2≤n≤5 and α>n−26−n, and under an additional assumption on ϕ′, it is furthermore asserted that the flux ∇(uϕ(v)) belongs to some reflexive Lebesgue space, and that (⋆) is satisfied in a standard weak sense.Finally, in the one-dimensional case a statement on global classical solvability is derived under the mere condition that ϕ∈C3((0,∞)) be positive.

Blow-up prevention by logistic source an N-dimensional parabolic-elliptic predator-prey system with indirect pursuit-evasion interaction

This paper is concerned with a pursuit-evasion model(1.0){ut=Δu−χ∇⋅(u∇w)+u(λ1−μ1ur1−1+av),x∈Ω,t>0,vt=Δv+ξ∇⋅(v∇z)+v(λ2−μ2vr2−1−bu),x∈Ω,t>0,0=Δw−w+v,x∈Ω,t>0,0=Δz−z+u,x∈Ω,t>0 in a smooth bounded domain Ω⊂RN,N≥1, where χ>0,ξ>0,λi>0,μi>0 and ri>1(i=1,2). For the case r1>1,r2>1, it will be proved that if (r1−1)(r2−1)>(N−2)+N, then for all appropriately regular nonnegative initial data u0 and v0, the problem possesses a unique global-in-time classical solution that is bounded in Ω×(0,∞). This extends the previous global boundedness result for r1=r2=2 and N≤3 (see Li et al. (2020) [12]).

A blow-up result for attraction-repulsion system with nonlinear signal production and generalized logistic source

This paper is devoted to studying the following chemotaxis system{ut=∇⋅(φ(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇ϖ)+λu(1−uα),x∈Ω,t>0,0=Δv−m1(t)+f1(u),m1(t)=1|Ω|∫Ωf1(u),x∈Ω,t>0,0=Δϖ−m2(t)+f2(u),m2(t)=1|Ω|∫Ωf2(u),x∈Ω,t>0,∂u∂ν=∂v∂ν=∂ϖ∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),x∈Ω, where Ω=BR(0)⊂Rn(n≥2) with R>0, ν denotes the outward unit normal vector on ∂Ω, χ,ξ,λ,α are positive constants, and φ(u),f1(u) and f2(u) are suitably regular functions satisfying φ(u)≤C0(1+u)−m, f1(u)≥k1(u+1)γ1 and f2(u)≤k2(u+1)γ2 for all u≥0 with C0,k1,k2,γ1,γ2>0 and m∈R. It is proved that if γ1>max⁡{α,γ2} and{γ1>2n(α+1),ifm≥0,γ1+m>2n(α+1),ifm<0, then there exists a suitable initial data u0 such that the corresponding solution (u,v,ϖ) of the system blows up in finite time. The results of this paper extend the blow-up criteria established in Liu et al. (2021) [28].

Existence, uniqueness and L∞-bound for weak solutions of a time fractional Keller-Segel system

We study the global existence, uniqueness and L∞-bound for the weak solutions to a time fractional Keller-Segel systems with logistic source∂αu∂tα=Δu−∇⋅u∇v+ua−bu,x∈ℝn,t>00=Δv+u,x∈ℝn,t>0where α ∈ (0,1), a ≥ 0, b > 0 with u(x,0) = u0, v(x,t) is represented by the Newton potentialvxt=1nn−2ωn∫ℝn1x−yn−2uydyWe divide the damping coefficient into different cases and use different methods to prove the existence of weak solutions: (i) when b>1−2n, for any initial value u0 and birth rate a ≥ 0, weak solutions exist globally. (ii) when 0<b≤1−2n, weak solutions have global existence under the condition of small initial data u0 and small birth rate a. Furthermore, by establishing fractional differential inequalities, the L∞-bound of weak solutions is obtained. Finally, we also prove that the weak solution must be unique when the damping effect is strong.

The Korteweg–de Vries equation on the half-line with Robin and Neumann data in low regularity spaces

The well-posedness of the initial–boundary value problem (ibvp) for the Korteweg–de Vries equation on the half-line is studied for initial data u0(x) in spatial Sobolev spaces Hs(0,∞), s>−3/4, and Robin and Neumann boundary data φ(t) in the temporal Sobolev spaces suggested by the time regularity of the Cauchy problem for the corresponding linear equation. First, linear estimates in Bourgain spaces are derived by utilizing the Fokas solution formula of the ibvp for the forced linear equation. Then, using these and the needed bilinear estimates, it is shown that the iteration map defined by the Fokas solution formula is a contraction in an appropriate solution space.

Global existence results for solutions of general conservative advection-diffusion equations in [formula omitted

In this paper a rigorous study concerning global existence results and estimates for the sup norm of non-negative bounded solutions for one-dimensional advection-diffusion equations ut+(b(x,t)uk+1)x=μ(t)uxx, with 0≤k<2 and initial data u0∈L1(R)∩L∞(R) is provided using a technique based on energy methods. In respect of the arbitrary advective speed term, it is only assumed that b(x,t) and ∂b(x,t)/∂x are limited.

Global boundedness and large time behavior of solutions to a chemotaxis system with flux limitation

This paper deals with the following cross-diffusion system{ut=∇⋅((u+1)m−1∇u)−∇⋅(u∇v(1+|∇v|2)α),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0, in a bounded domain Ω⊂Rn, n≥2 with smooth boundary. In this framework, it is shown that the problem possesses a unique global bounded classical solution under the condition that α>2n−2−mn2(n−1) and m≥1. Moreover, it is asserted that the corresponding solution exponentially converges to the constant stationary solution (u0‾,u0‾) when the initial data u0 is sufficiently small, where u0‾:=∫Ωu0|Ω|.

Ill-posedness for the Cauchy problem of the Camassa-Holm equation in [formula omitted

For the famous Camassa-Holm equation, the well-posedness in Bp,11+1p(R) with p∈[1,∞) and the ill-posedness in Bp,r1+1p(R) with p∈[1,∞],r∈(1,∞] had been studied in [13,14,16,23], that is to say, it only left an open problem in the critical case B∞,11(R) proposed by Danchin in [13,14]. In this paper, we solve this problem by proving the norm inflation and hence the ill-posedness for the Camassa-Holm equation in B∞,11(R). Therefore, the well-posedness and ill-posedness for the Camassa-Holm equation in all critical Besov spaces Bp,11+1p(R) with p∈[1,∞] have been completed. Finally, since the norm inflation occurs by choosing an special initial data u0∈B∞,11(R) but u0x2∉B∞,10(R) (an example implies B∞,10(R) is not a Banach algebra), we then prove that this condition is necessary. That is, if u0x2∈B∞,10(R) holds, then the Camassa-Holm equation has a unique solution u(t,x)∈CT(B∞,11(R))∩CT1(B∞,10(R)) and the norm inflation will not occur.

Global existence and boundedness of solutions to a parabolic attraction-repulsion chemotaxis system in [formula omitted]: The repulsive dominant case

This paper deals with the initial value problem for a parabolic attraction-repulsion chemotaxis system in R2:{∂tu=Δu−∇⋅(u∇(β1v1−β2v2)),t>0,x∈R2,∂tvj=Δvj−λjvj+u,t>0,x∈R2(j=1,2),u(0,x)=u0(x),vj(0,x)=vj0(x),x∈R2(j=1,2) with positive parameters β1, β2, λ1, λ2 and nonnegative initial data u0, v10, v20. It is well known that the sign of β1−β2 and the mass on u0 play a crucial role in the global boundedness of the system. Specifically, in the attractive dominant case β1>β2, the uniform boundedness of nonnegative solutions has been guaranteed under the condition ∫R2u0dx<4π. Also, in the balance case β1=β2, it has been proven that every nonnegative solution is bounded uniformly in time without any restriction on the mass u0. In this paper, we discuss the global existence and boundedness of nonnegative solutions to the system in the repulsive dominant case β1<β2.

On the local existence for Hardy parabolic equations with singular initial data

We consider the singular nonlinear equation ut−Δu=|⋅|−γf(u) in Ω×(0,T) with γ>0 and Dirichlet conditions on the boundary. This equation is known in the literature as a Hardy parabolic equation. The function f:[0,∞)→[0,∞) is continuous and non-decreasing, and Ω is either a smooth bounded domain containing the origin or the whole space RN. We determine necessary and sufficient conditions for the existence and non-existence of solutions for initial data u0∈Lr(Ω),u0≥0, with 1≤r<∞. We also give a uniqueness result.

Existence and global asymptotic stability in a fractional double parabolic chemotaxis system with logistic source

This paper studies a double parabolic chemotaxis system with logistic source and a fractional diffusion of order α∈(0,2)ut=−Λαu−χ∇⋅(u∇v)+au−bu2,vt=Δv−v+uon two dimensional periodic torus T2. In contrast to the well-known Neumann heat semigroup {etΔ}t≥0 estimates in a smoothly bounded domain Ω⊂Rn introduced by Winkler (2010), we obtain the spatio-temporal estimates of the analytic semigroup {Ttα(x)}t≥0 and {Tt(x)}t≥0 which are generated by −(−Δ)α2−I and Δ−I respectively over periodic torus T2. With the help of these conclusions, we can use the semigroup method to study the global existence and the asymptotic behavior of the above fractional chemotaxis model, which has not been studied yet. It is proved that for any nonnegative initial data (u0,v0)∈H4(T2)×H5(T2), if 1<α<2, there admits a unique globally classical solution. Furthermore, if a=0, we can obtain that the solution components u and v converge to zero with respect to the norm in L∞(T2) as t→∞.

Improvement of conditions for boundedness in a chemotaxis consumption system with density-dependent motility

This work considers the chemotaxis model with density-dependent motility ut=Δ(ϕ(v)u),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R2, with the nonnegative initial data (u0,v0)∈[W1,∞(Ω)]2. The motility function ϕ(v) satisfies ϕ(v)∈C3([0,∞)) with ϕ(v)>0 and ϕ′(v)<0 for all v≥0. For all suitably regular initial data, we prove that this system has a unique global bounded classical solution. The purpose of this work is to remove the smallness assumption on ‖v0‖L∞(Ω) in Li and Zhao (2021).

On nonnegative solutions for the Functionalized Cahn–Hilliard equation with degenerate mobility

The Functionalized Cahn–Hilliard equation has been proposed as a model for the interfacial energy of phase-separated mixtures of amphiphilic molecules. We study the existence of a nonnegative weak solutions of a gradient flow of the Functionalized Cahn–Hilliard equation subject to a degenerate mobility M(u) that is zero for u≤0. Assuming the initial data u0(x) is positive, we construct a weak solution as the limit of solutions corresponding to non-degenerate mobilities and verify that it satisfies an energy dissipation inequality. Our approach is a combination of Galerkin approximation, energy estimates, and weak convergence methods.

Global large solution for the tropical climate model with diffusion

We study the d-dimensional (d=2,3) tropical climate model with only the dissipation of the first baroclinic model of the velocity −ηΔv. By choosing a class of special initial data (u0,v0,𝜃0) whose Hs(ℝd) norm can be arbitrarily large, we obtain the global smooth solution of d-dimensional tropical climate model.

Global large solution for the tropical climate model with diffusion

We study the d-dimensional (d=2,3) tropical climate model with only the dissipation of the first baroclinic model of the velocity −ηΔv. By choosing a class of special initial data (u0,v0,𝜃0) whose Hs(ℝd) norm can be arbitrarily large, we obtain the global smooth solution of d-dimensional tropical climate model.

Local Existence and Non-Existence for a Fractional Reaction-Diffusion Equation in Lebesgue Spaces

Abstract We consider the following fractional reaction-diffusion equation u t ( t ) + ∂ t ∫ 0 t g α ( s ) A u ( t − s ) d s = t γ f ( u ) , $$ u_t(t) + \partial_t \int\nolimits_{0}^{t} g_{\alpha}(s) \mathcal{A} u(t-s) ds = t^{\gamma} f(u),$$ where g α (t) = t α−1/Γ(α) (0 &lt; α &lt; 1), f ∈ C([0, ∞)) is a non-decreasing function, γ &gt; −1, and A $\mathcal{A}$ is an elliptic operator whose fundamental solution of its associated parabolic equation has Gaussian lower and upper bounds. We characterize the behavior of the functions f so that the above fractional reaction-diffusion equation has a bounded local solution in L r (Ω), for non-negative initial data u 0 ∈ L r (Ω), when r &gt; 1 and Ω ⊂ ℝ N is either a smooth bounded domain or the whole space ℝ N . The case r = 1 is also studied.

Energy decay for a viscoelastic wave equation with space-time potential in [formula omitted

In this paper, we consider the following viscoelastic wave equation{utt−(Δu−∫0tg(t−s)Δu(s)ds)+b(t,x)ut+|u|p−1u=0,t>0,x∈Rnu(0,x)=u0(x),ut(0,x)=u1(x),x∈Rn with space-time dependent potential and where the initial data u0(x), u1(x) have compact support. Under suitable assumptions on the relaxation function g and the potential b, we obtain a more general energy decay result using the perturbed energy method.

Boundedness in a chemotaxis–haptotaxis model with gradient-dependent flux limitation

This paper deals with a chemotaxis–haptotaxis system with gradient-dependent flux-limitation ut=Δu−χ∇⋅(uf(|∇v|2)∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0,under a smooth bounded domain Ω⊂Rn,n∈{2,3}, where χ, ξ and μ are positive parameters, f∈C2([0,∞)) satisfies the condition f(|∇v|2)≤(1+(|∇v|2))p−22,with 1<p<nn−1. It is proved that for sufficiently smooth initial data (u0,v0,w0), the corresponding initial–boundary problem possesses a unique classical solution, which is uniformly bounded in time.